Polynomials Overview

Polynomials

Definition: A polynomial is a mathematical expression composed of variables, coefficients, and non-negative integer exponents.
General Form:
- A polynomial in one variable (x) can be expressed as:
  - ( P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 )
- Where ( a_i ) are coefficients, ( n ) is a non-negative integer (degree), and ( a_n \neq 0 ).
Types of Polynomials:
- Monomial: A polynomial with one term (e.g., ( 3x^2 )).
- Binomial: A polynomial with two terms (e.g., ( 2x + 5 )).
- Trinomial: A polynomial with three terms (e.g., ( x^2 + 3x + 2 )).
- Polynomial Degree: The highest exponent of the variable is considered the degree of the polynomial.
  - Example: Degree of ( 4x^3 + 2x^2 + 7 ) is 3.
Operations with Polynomials:
- Addition/Subtraction: Combine like terms.
  - Example: ( (2x^2 + 3x) + (4x^2 + x) = 6x^2 + 4x ).
- Multiplication: Use the distributive property (FOIL for binomials).
  - Example: ( (x + 2)(x + 3) = x^2 + 5x + 6 ).
- Division: Can use long division or synthetic division.
Factoring Polynomials:
- Common Techniques:
  - Factoring out the greatest common factor (GCF).
  - Factoring by grouping.
  - Using special product formulas (e.g., difference of squares).
- Quadratic Polynomials: Can often be factored into the form ( (ax + b)(cx + d) ) for ( ax^2 + bx + c ).
Roots/Zeros of Polynomials:
- Roots of a polynomial are the values of ( x ) that make ( P(x) = 0 ).
- Methods to find roots include:
  - Graphing.
  - Factoring.
  - Applying the quadratic formula for quadratics:
    - ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
Graphing Polynomials:
- The shape of the graph (end behavior) is influenced by the degree and leading coefficient.
- A polynomial of degree ( n ) has up to ( n ) roots.
Common Polynomial Forms:
- Standard Form: Written as ( a_n x^n + a_{n-1} x^{n-1} + ... + a_0 ).
- Factored Form: Written as products of linear factors (e.g., ( (x - r_1)(x - r_2)...(x - r_n) )).
Important Concepts:
- Leading Coefficient: The coefficient of the term with the highest degree.
- End Behavior: Describes how the polynomial behaves as ( x ) approaches positive or negative infinity.
- Symmetry: Polynomials can be even (symmetric about the y-axis) or odd (symmetric about the origin).

Polynomials Overview

A polynomial is an expression consisting of variables, coefficients, and exponents, with exponents being non-negative integers.
General form for a polynomial in one variable ( x ) is expressed as ( P(x) = a_n x^n + a_{n-1} x^{n-1} +...+ a_1 x + a_0 ).
Coefficients denoted as ( a_i ), degree ( n ) must be a non-negative integer, and ( a_n ) cannot be zero.

Types of Polynomials

Monomial: Contains a single term (e.g., ( 3x^2 )).
Binomial: Contains two terms (e.g., ( 2x + 5 )).
Trinomial: Contains three terms (e.g., ( x^2 + 3x + 2 )).
The degree of a polynomial is determined by the highest exponent of its variable.

Polynomial Operations

Addition/Subtraction: Requires combining like terms (e.g., ( (2x^2 + 3x) + (4x^2 + x) = 6x^2 + 4x )).
Multiplication: Apply the distributive property; for binomials, use FOIL. Example: ( (x + 2)(x + 3) = x^2 + 5x + 6 ).
Division: Can be conducted through long division or synthetic division.

Factoring Polynomials

Common techniques include factoring out the greatest common factor (GCF), grouping, and applying special product formulas, such as the difference of squares.
Quadratic polynomials can often be factored into the form ( (ax + b)(cx + d) ) for expressions like ( ax^2 + bx + c ).

Roots and Zeros

Roots are the values of ( x ) that satisfy ( P(x) = 0 ).
Methods for finding roots include graphing, factoring, and using the quadratic formula for quadratics: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

Graphing Polynomials

Graph shape (end behavior) is determined by the degree and leading coefficient.
A polynomial of degree ( n ) can have up to ( n ) roots.

Polynomial Forms

Standard Form: Written as ( a_n x^n + a_{n-1} x^{n-1} +...+ a_0 ).
Factored Form: Expressed as products of linear factors (e.g., ( (x - r_1)(x - r_2)...(x - r_n) )).

Important Concepts

Leading Coefficient: Refers to the coefficient of the term with the highest degree.
End Behavior: Describes polynomial behavior as ( x ) approaches positive or negative infinity.
Symmetry: Polynomials may display symmetry; even polynomials are symmetric about the y-axis while odd polynomials are symmetric about the origin.